ap
year. Denoting, therefore, the number of the century (or the date after the
two right-hand digits have been struck out) by c, the value of L must be
increased by 10 + (c - 16) - ((c - 16) / 4)_w . We have then

L = 7m + 3 - x - (x/4)_w + 10 + (c - 16) - ((c - 16) / 4)_w;

that is, since 3 + 10 = 13 or 6 (the 7 days being rejected, as they do not
affect the value of L),

L = 7m + 6 - x - (x/4)_w + (c - 16) - ((c - 16) / 4)_w.

x = 1839, (x/4)_w = (1839/4)_w = 459, c = 18, c - 16 = 2,
and ((c - 16) / 4)_w = 0.

((26 + 11(N - 6)) / 30)_r. But the numerator of this fraction becomes by
reduction 11 N - 40 or 11 N - 10 (the 30 being rejected, as the remainder
only is sought) = N + 10(N - 1); therefore, ultimately,

J = ((N + 10(N - 1)) / 30)_r.

On account of the solar equation S, the epact J must be diminished by unity
every centesimal year, excepting always the fourth. After x centuries,
therefore, it must be diminished by x - (x/4)_w. Now, as 1600 was a leap
year, the first correction of the Julian intercalation took place in 1700;
hence, taking c to denote the number of the century as before, the
correction becomes (c - 16) - ((c - 16) / 4)_w, which [v.04 p.0999] must be
deducted from J. We have therefore

S = - (c - 16) + ((c - 16) / 4)_w.

With regard to the lunar equation M, we have already stated that in the
Gregorian calendar the epacts are increased by unity at the end of every
period of 300 years seven times successively, and then the increase takes
place once at the end of 400 years. This gives eight to be added in a
period of twenty-five centuries, and x/25 in x centuries. But 8x/25 = 1/3
(x - x/25). Now, from the manner in which the intercalation is directed to
be made (namely, seven times successively at the end of 300 years, and once
at the end of 400), it is evident that the fraction x/25 must amount to
unity when the number of centuries amounts to twenty-four. In like manner,
when the number of centuries is 24 + 25 = 49, we must have x/25 = 2; when
the number of centuries is 24 + 2 x 25 = 74, then x/25 = 3; and, generally,
when the number of centuries is 24 + n x 25, then x/25 = n + 1. Now this is
a condition which will evidently be expressed in general by the formula n -
((n + 1) / 25)_w. Hence the correction of the epact, or the number of days
to be intercalated after x centuries reckoned from the commencement of one
of the periods of twenty-five centuries, is {(x - ((x+1) / 25)_w)